One issue is the series resistence. You might want to keep passive components out of the signal path and this would possibly increase signal purity and lower the insertion loss of the network. IE allow a little mor power to the driver.

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If I want to deal with a cone breakup hump above a woofers crossover frequency, I like to use a parallel notch filter. The parallel filter needs the series impedance (which the main crossover inductor is providing) or else it wont work.

If the notch filter is the only crossover part on the driver, a series will have to do otherwise (for a parallel to work) you'd need to add a series resistor to take the energy within the filter band.

All this is very generalised. I find filters are situation specific but I tend to prefer the parallel where possible. One reason might be that the drivers seem an easier impedance target to work with than the entire crossed over segment (if you look at where the respective filter's effects lie)

Indm's usage is in conflict with how the original poster defined series/parallel.

The main difference is in whether or not you can tolerate a rise in impedance (parallel notch) or a dip in impedance (series notch) The parallel filter will also have an insertion loss below the filter frequency due to the inductor DCR.

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Thanks Ron E, to clarify I refered to the parallel notch filter as the series string of components together placed in parallel with the driver, and the series notch filter as the parallel(ed) components together placed in series with the driver.

So, as I understand:
A serie notch filter in parallel with the driver is used for cone breakup. Cone breakup is mostly in the take-over region between woofer and tweeter. Both must be in phase. Is there no risk that the signals are not in phase anymore after inserting the notch filter?

A parallel notch filter (with components of the notch filter in parallel) is used in the passband of the crossover filter. Am I right?

I still don't see the reason why a parallel notch filter is better in the passband of the crossover? Can a serial notch not do the work.

One more question:
Why is a serial notch filter always used refered to the negative to remove bumps?
Is it not possible to use a serial notch filter to remove dips in the frequency when the filter is refered to the positive amplifier point.

I mean one side of the filter on the speaker + and one side on the amplifier + to remove dips in FR. At resonance frequentie of the notch impedance is zero and the speaker gets more energy from the amplifier to remove the dip in FR.